To find the total area of the box net, we'll decompose it into simpler shapes and calculate their individual areas before summing them up. The box net includes rectangles, squares, and trapezoids.
Here's a detailed breakdown:
### Central Rectangle:
This central rectangle has a width of 3 inches and a length of 12 inches (since 3 inches + 3 inches + 6 inches = 12 inches). The area is:
\[ \text{Area of central rectangle} = \text{length} \times \text{width} = 12 \, \text{inches} \times 3 \, \text{inches} = 36 \, \text{square inches} \]
### Squares:
There are four squares, each with a side length of 3 inches. The area of one square is:
\[ \text{Area of one square} = \text{side}^2 = 3 \, \text{inches} \times 3 \, \text{inches} = 9 \, \text{square inches} \]
Since there are four squares:
\[ \text{Total area of squares} = 4 \times 9 \, \text{square inches} = 36 \, \text{square inches} \]
### Trapezoids:
Each trapezoid has a longer base of 3 inches, a shorter base of 1.2 inches, and a height of 1 inch. The area of one trapezoid is calculated using the trapezoid area formula:
\[ \text{Area of one trapezoid} = \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height} \]
\[ \text{Base}_1 = 3 \, \text{inches}, \, \text{Base}_2 = 1.2 \, \text{inches}, \, \text{Height} = 1 \, \text{inch} \]
\[ \text{Area of one trapezoid} = \frac{1}{2} \times (3 + 1.2) \times 1 = \frac{1}{2} \times 4.2 \times 1 = 2.1 \, \text{square inches} \]
Since there are four trapezoids:
\[ \text{Total area of trapezoids} = 4 \times 2.1 \, \text{square inches} = 8.4 \, \text{square inches} \]
### Total Area:
Summing up the areas of all shapes, we get the total area of the cardboard needed:
\[ \text{Total area} = \text{Area of central rectangle} + \text{Area of squares} + \text{Area of trapezoids} \]
\[ \text{Total area} = 36 \, \text{square inches} + 36 \, \text{square inches} + 8.4 \, \text{square inches} \]
\[ \text{Total area} = 80.4 \, \text{square inches} \]
So, the total amount of cardboard needed to make the box is:
\[ \boxed{80.4 \, \text{square inches}} \]
Area and Volume Unit Test
7 of 147 of 14 Items
Question
Use the image to answer the question.
An illustration shows an unfolded box net. It is comprised of a rectangle at the center, two squares extending into three trapezoids on the top and bottom, and a trapezoid on the left of the rectangle. The top side of the rectangle is labeled in three distinct sections: starting at the left, a solid line of 3 is drawn, followed by a dashed line of 3, and then a solid line of 6. The bottom of the rectangle is labeled in three distinct sections: a solid line of 6, a dashed line of 3, and a solid line of 3. The left width of the rectangle is drawn as a dashed line, labeled as 3, which becomes the base of a shaded trapezoid. The dashed lines on the top and bottom of the rectangle are extended into squares of side 3, sides drawn in dashed lines. The three outer sides of each square become the longer bases of shaded trapezoids. The top shorter side of the trapezoid is labeled as 1.2 and the perpendicular height is 1.
Decompose the box net into simple polygons to find how much cardboard in square inches is needed to make the cube. Assume that all angles that look like right angles are right angles. All polygons that look congruent are congruent. What is the area of the shape in square inches?
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